Let $E$ be a normed (real) space which is not complete. Is it always possible to find $f$ a continuous bijective linear function from $E$ to $E$ such that $f^{1}$ is not continuous?
No. If $E$ has codimension less than the continuum in a Banach space then such an $f$ must be open. This was proved by Saxon and Levin; see the proposition on page 95 of which is Saxon, Stephen; Levin, Mark, Every countablecodimensional subspace of a barrelled space is barrelled. Proc. Amer. Math. Soc. 29 1971 91–96. So for an example, take any discontinuous linear functional on a Banach space and let $E$ be its kernel. 

